In this chapter we turn to the inverse scattering problem. There are in fact several possible inverse problems, but this thesis is only concerned with the following one:
given far field patterns for all incident directions with a fixed wavenumber, find the shape of the sound-hard obstacle. This problem turns out to be nonlinear and ill-posed. The ill-posedness makes it a challenging problem especially from a numerical point of view. We begin this chapter with a brief review of the properties of this inverse problem. Then in the second section we study a relatively new method for solving inverse scattering problems. The method is known as factorization method.
3.1 The inverse problem
In addition to being inverse to a problem called direct problem, an inverse problem is (typically) ill-posed. According to Hadamard’s classical definition a problem is well-posed if
(i) it has a solution,
(ii) the solution is unique, and
(iii) the solution is stable, i.e., it depends continuously on the data.
A problem that fails to satisfy at least one of these conditions is said to be ill-posed.
The inverse scattering problem is ill-posed because it does not satisfy condition (iii) and in practical applications there may be problems with condition (i) also.
Let us now precisely formulate the inverse problem considered in this work:
The inverse problem: Given the far field patternFD(ϕ;d, k) for allϕ, d∈S1 and fixed k >0, determine the shape of the sound-hard obstacle D.
The rest of this section is devoted to considering this problem with respect to con-ditions (i)–(iii).
Assuming that the given data represents far field patterns of some obstacle, there clearly exists a solution. However, real-world measurements as well as numerical computations always contain errors, and hence it may happen that the given mea-surement data does not represent far field patterns, in which case the existence condition (i) is violated.
There are many uniqueness results for inverse scattering problems. The following theorem guarantees existence of a unique solution to our inverse problem.
Theorem 3.1.1. LetD1 andD2 be two sound-hard scatterers whose far field patterns coincide for all incident directions and a fixed wave number. ThenD1 =D2. Proof. See [12, Theorem 3.1.1].
This result verifies, at least in theory, that the knowledge of the far field patterns for all incident plane waves with a fixed wave number suffices to determine the obstacle uniquely, and in this sense the inverse problem is a reasonable problem from practical point of view.
Finally we consider the stability condition (iii). The inverse scattering problem begins from the knowledge of the far field patterns. To see the ill-posedness of the inverse problem we have to consider the mappingws 7→FD from the scattered field ws to the far field patternFD defined by (2.53). This is due to the fact that the aim in the inverse problem is, in a sense, to recover the scattered field from the knowledge of the far field patterns: reconstructing the obstacle is equivalent to determining the zeros of the normal derivative∂w/∂ν of total field w=wi+ws where the incident field wi is known.
To rigorously verify the ill-posedness of the inverse problem we could use func-tion series representafunc-tions for scattered fields and for far field patterns or exploit functional analytic results for compact operators. Here we just state that the ill-posedness is caused by the smoothing effect of the integration in (2.53).
3.2 The factorization method
The factorization method is a relatively new method for solving shape identifica-tion problems related to inverse problems such as inverse scattering problems and electrical impedance tomography. It was developed by Andreas Kirsch and Natalia Grinberg. Detailed information and analysis, as well as references to the original publications, can be found in their recent monograph [9]. Here we just briefly outline the derivation of the method.
The factorization method (and its name) is based on a factorization of the far field operatorF :L2(S1)→L2(S1) defined by
(F g)(ϕ) = Z
S1
FD(ϕ;d, k)g(d)ds(d), ϕ∈S1.
Notice that this operator contains all the information given in the far field patterns.
The operatorF is compact and has a factorization of the form F =GT G∗,
whereGandG∗are compact operators andT is an isomorphism between appropriate spaces. The fundamental result is that a point z ∈R2 belongs to the obstacle D if and only if the functionφz ∈L2(S1), given by
φz(ϕ) =e−ikϕ·z,
belongs to the range ofG. This result is not very useful from computational point of view; however, a computationally attractive formulation can be achieved as follows.
In the case of sound-hard obstacles (Neumann boundary conditions) the far field operatorF can be shown to be normal, that is,
F∗F =F F∗,
where the operator F∗ is the L2-adjoint of F. Hence, from the spectral theory of normal operators we know thatF can be represented as
F g = X∞
j=1
λj(g, ψj)ψj,
where (·,·) denotes theL2inner product, andλj ∈C,j = 1,2, . . .are the eigenvalues of F with the corresponding eigenfunctions ψj, j = 1,2, . . .. Moreover, it can be shown that the ranges of operatorsG and (F∗F)1/4 coincide, where
(F∗F)1/4g = X∞
j=1
q
|λj|(g, ψj)ψj.
Thus z ∈R2 belongs to D if and only if there exists g ∈L2(S1) such that
(F∗F)1/4g =φz. (3.1)
Writingφz =P∞
j=1(φz, ψj)ψj and applying Picard’s criterion we conclude that (3.1) is solvable if and only if
X∞
j=1
|(φz, ψj)|2
|λj| (3.2)
converges. The main result can now be formulated as follows.
Theorem 3.2.1. Assume that k2 is not a Neumann eigenvalue of −∆ in D, i.e., there exists no nontrivial solutionw∈C2(D)∩C1(D)to the Helmholtz equation such that ∂w/∂ν = 0 on ∂D. Then z ∈R2 belongs to D if and only if (3.2) converges,
that is,
W(z) :=
X∞
j=1
|(φz, ψj)|2
|λj|
!−1
>0. (3.3)